In this article, we summarise the rotation-free and quaternionic parametrisation of a rigid body. We derive and explain the close interrelations between both parametrisations. The internal constraints due to the redundancies in the parametrisations, which lead to DAEs, are handled with the null space technique. We treat both single rigid bodies and general multibody systems with joints, which lead to external joint constraints. Several numerical examples compare both formalisms to the index reduced versions of the corresponding standard formulations.